David Marain over at MathNotations has been periodically tweeting an SAT problem of the day via @dmarain. A recent series of tweets asked:

A 42 oz mix of nuts is 6 parts peanuts to 1 part cashews. How many ounces of cashews must be added to make a make the mixture 2 parts cashews to 1 part peanuts? Does Singapore (bar) model method work here?

Sure it can! Here’s one solution.

First model the beginning mixture at 6:1.

Next, we can figure out how many units need to be added to the cashews to make the cashews to peanuts ratio 2:1.

Since there are 11 additional units and the value of each unit is 6 oz, we can find the value of the added cashews.

66 ounces of cashews must be added to the peanuts to make the mixture 2:1.
Changing Ratios is taught in level 6A in the Primary Mathematics materials (U.S. Edition), typically used in 6th grade classrooms.

Out in Left Field posts a Math Problem of the Week comparing different curricula that schools use. Last week’s Riddles in 2nd Grade Investigations vs. Singapore Math and the ensuing comments brought up discussion on the value of writing in the mathematics classroom.  I picked up a book entitled: Journal Writing in the Mathematics Classroom (Primary) when I was in Singapore. It is written and published by professors at Singapore’s National Institute of Education.

Among the chapters listed:

Why use Journal Writing?

Journal writing reinforces the learning and provides pupils with opportunities to engage in reflection, question their own understanding, connect the abstract and the concrete, and apply the knowledge they have acquired to solve problems.

How to Carry Out Journal Writing?

This section starts with an this powerful statement about journal writing in the classroom:

Journal writing is a complex process that requires effort and patience.

The authors further suggest that these open-ended prompts encourage pupils to write about their opinions and feelings on mathematics. They have adapted and describe three types of general writing prompts:

1. Affective or Attitudinal (How do you feel?)

• My best kept secret about math is …
• If math could be a colour (shape, sound) it would be … because

2. Mathematical content (What is it about?)

• How would you describe a …
• What patterns do you notice in …

3. Process (Explain how!)

• Find something that you learned today that is similar to something you already knew.
• You know several ways to … Which method is you favourite? Why?

There is a list of 15 of each type of general writing prompt. The bulk of the book, however, focuses on specific writing prompts that are based on mathematical topics.

Possible Negative Aspects of Journal Writing

(Lessons learned the hard way when I was teaching!)

a. The potential for the teacher to hurt pupil’s feelings.
b. The loss of instructional time to teach syllabuses
c. Tremendous increase in the marking load of the teacher.
d. Emphasis on language proficiency

Scoring Rubrics and Student Examples

This section includes examples of two types of scoring rubrics: Analytic, which allows for separate evaluation of selected factors and Holistic, which can be used when teachers want to rate student responses more generally.

A Collection of Specific Writing Prompts

Finally, there are 55 specific writing prompts differentiated by grade level and topic. Topics include: Whole Numbers, Fractions, Decimals, Percentage, Ratio, Rate, Measurement, Geometry, Statistics and Algebra. Some examples:

Topic: Whole Numbers
Level: Primary 2 – 6

Write a word problem and make a picture that goes with 4 x 3.

Topic: Decimals
Level: Primary 4 – 6

Find two decimal numbers between 0.2 and 0.3. How many decimal numbers are there between 0.2 and 0.3? Explain.

From publisher Frank Schaffer and Singapore Asian Publications comes a series of books: Singapore Math: 70 Must-Know Word Problems. From the back cover:

This book is designed to help students master word problems, which are often tricky and challenging…This book is perfect for students familiar with Singapore Math and for those who just need extra practice with word problems.

The 70 Must-Know Word Problems books are marked by levels that are considered a grade above for the U.S. market. For example, the Level 6 book claims to be appropriate for students in grade 7 and the Level 4 book is listed as appropriate for students in grade 5. If your child has been learning with Primary Mathematics and you’d like to pick up one of these workbooks, you should probably pick up the level that correlates to their current grade. Meaning, if your child is currently working their way through the Level 4 Primary Mathematics textbook, then the level 4 70 Must-Know Word Problems book will work as a supplement. The problems in the books are extra practice type problems, designed to give students similar to the ones in the Primary Mathematics textbooks. If you student is look for something harder, you might want to consider the Challenging Word Problems for Primary Mathematics series.

The 70 Must-Know series does not follow the Primary Mathematics lesson progression, the questions jump between concepts. Questions #3 and #47 in the Level 6 book, for example, are both on Volume. Question #2 & 5 cover the four operations with money  and Question #4 is on percentages, so you may need to verify that a topic has been covered in class if your planning on using the book at home. The books have little instruction, however they do include a four page Introduction to Singapore Math (be forewarned, it’s written in six-point font!). Each problem has its own full page for work and there are fully worked solutions included in the answer key.

Here’s Question #70 from the Level 4 book. It’s listed in the table of contents under “Addition, Subtraction, Multiplication and Division of Whole Numbers”.

At a movie, 1/4 of the people in the theater were men, 5/8 were women and the rest were children. If there were 100 more women than children, what was the total number of people in the theater?

And Question # 48 from Level 6:

A box contained some red, blue, and green markers. For every 5 red markers, there were 2 blue markers. For every 3 blue markers, there were 5 green markers.

(a) Find the ratio of red markers to blue markers to green markers.

(b) When 6 red markers were removed from the box, 3/7 of the remaining markers were red markers. How many markers were left in the box?

Singapore Math: 70 Must-Know Word Problems was$12.99 at my local Barnes and Noble and Borders stores. The easy accessibility of these books make them great for parents looking for additional problems for their students. Teachers looking to use these materials in class need to keep on mind that these books may already be in their students’ homes!

Google Books is offering free previews of books in this series:

• Singapore Math 70 Must-Know Word Problems, Level 1
• Singapore Math 70 Must Know Word Problems , Level 2
• Singapore Math 70 Must-Know Word Problems, Level 3
• Singapore Math 70 Must-Know Word Problems, Level 5

In late April, two mathematics conferences were held in San Diego: The NCSM (National Council of Supervisors of Mathematics) and the NCTM (National Council of Teachers of Mathematics). This is the final in a series of posts from those conferences.

NCTM Session:

• Does Singapore Mathematics (SM) Enhance Students’ Learning in the United States? By Richard Askey, Patsy Wang- Iverson and Ban-Har Yeap

Here is the entire session description from the NCTM Program Handbook:

Many teachers want to use SM with their students, but they have been unable to provide data on its effectiveness in the United States. The speakers will report results of a longitudinal study of SM implementation, offer examples of concepts student learn and offer ways in which all students can learn math to high levels.

You can find the slides from Dr. Yeap on his website. After a quick review of the educational system in Singapore, he discussed the key components of their primary school mathematics program.

• Emphasis on problem solving
• Enrichment for all students
• Parental expectations
• System-wide interventions
• Using a concrete – pictorial - abstract progression, especially the models
• Teacher education, both pre-service and professional development

Dr. Yeap presented, then quickly left in order to get to his next session on solving Singapore Math Problems that was a ten minute walk away in another venue. Dr. Askey took over and reviewed some of the problems at greater depth.

Dr. Wang-Iverson provided everyone with a copy of the American Educator magazine containing a recent and very informative article she co-authored: Beyond Singapore’s Mathematics Textbooks -Focused and Flexible Supports for Teaching and Learning.

At the close of the session, I was not the only attendee wondering what had happened to that study. The success of Singapore Math in the United States tends to be based on anecdotal evidence. Teachers will say, “I could really tell my students were ‘getting it’ “or “our test scores soared”. Compelling, but hardly objective evidence of the curriculum’s efficacy here in the United States.

Typically, NCTM sessions are submitted over one year in advance and the presenters had hoped for valid research out of the promoted study. Due to some issues with the study cited in the NCTM session description, it will not be published. This longitudinal data on Singapore Math in the United States has been hard to come by. While hundreds of schools have adopted the curriculum, long-term information simply isn’t available from any of the larger implementations, yet.

The What Works Clearinghouse a department of the U.S. Department of Education Institute of Education Sciences, has looked at studies of Singapore Math at the middle school and elementary levels. At the elementary level, as of July 2007, the organization found that “no studies meet eligibility screens”. The WWC released a report in April 2009 regarding the middle school level material for Singapore Math (New Elementary Mathematics) which determined:

No studies of Singapore Math that fall within the scope of the Middle School Math review protocol meet What Works Clearinghouse (WWC) evidence standards. The lack of studies meeting WWC evidence standards means that, at this time, the WWC is unable to draw any conclusions based on research about the effectiveness or ineffectiveness of Singapore Math.

Note that the WWC simply didn’t find any studies among the twelve they reviewed that satisfy their protocol, so they can’t draw any conclusions as to the effectiveness of the Singapore Math Middle School Curriculum.

I’ve heard Patsy Wang-Iverson sum it up succinctly:

Without data, the chatta don’t matta.

In late April, two mathematics conferences were held in San Diego: The NCSM (National Council of Supervisors of Mathematics) and the NCTM (National Council of Teachers of Mathematics). This is the fourth in a series of posts from those conferences.

NCSM Session:

• Integrating Curriculum, Assessment, and Teacher Professional Development: Singapore and the United States by Ban Har Yeap, Khoon Yoong Wong, Jeremy Roschelle and a colleague.

The final Singapore Math-related sessions at NCSM was the result of a joint research project between the National Institute of Education (NIE)  in Singapore and the Stanford Research Institute International Center for Technology in Learning in Menlo Park, California (SRI).

Dr. Wong spoke first about the teacher training provided in Singapore. The National Institute of Education is the sole teacher preparation facility in Singapore. This allows for standardized pre-service program. He shared the Pre-service Mathematics Teacher Education Framework that guides teacher training in Singapore:

(click to enlarge)

Teachers in Singapore take one of a couple of routes:

1. Diploma in Education (Dip Ed): 2 years, full-time, Primary.
2. Bachelor Degree: BA (Ed) or BSc (Ed): 4 years, full-time, Primary or Secondary.
3. Postgraduate Diploma in Education (PGDE): 1 year, full-time, Primary or Secondary.

Most interestingly, Dr. Wong added the number of courses out of the total courses that each pre-service track requires:
SK = Subject Knowledge (math for school teaching)
CK = Curriculum Studies (math pedagogy)

1. Diploma: SK =  6 /69    CK = 8 /69
2. Degree: SK = 4/126    CK = 10/126
3. PGDE (primary) SK = 4/44    CK =  8/44

Further reading on teacher preparation: Breaking the Cycle: An international comparison of U.S. mathematics teacher preparation.

Dr. Yeap then spoke about professional development for teachers in Singapore. Many people believe that Singaporean primary teachers are specialists, which they pointed out was simply not factual. From his presentation:

Regardless of where it is initiated, most professional development in Singapore has been workshop-style. One of the big challenges of this type of professional development is that there is no follow-up with the teachers on the material presented. Teachers attend the workshop, then go back to their classroom. (Sound familiar?) Singapore is now working on developing a professional learning culture through more reflective teacher practices. Newer avenues for professional development include Lesson Study, Action Research and Professional Learning Communities.

A major difference between United States and Singapore is the allotment of 100 hours of professional development each year for teachers. This statistic is commonly cited as something that could never be achieved in the United States. In actuality, 100 hours amounts to two and a half weeks of time. Additionally, in Singapore, these 100 hours can be spent on activities that enhance the teacher as a person. Calligraphy, school-based team meetings and pottery-making are among activities that might count for these 100 hours. When questioned, Dr. Yeap also qualified that teachers in Singapore might worry if one teacher was putting in more hours than they were. High expectations among the population dictate that the 100 hours is usually a minimum.

You can view Dr. Yeap’s entire presentation on his website, along with other presentations he has done around the world. Best quote from the session comes from him:

Mathematics is an excellent vehicle for the development and improvement of a person’s intellectual competence.

The researchers from SRI spoke next on the findings from their research project. In the NCSM Conference book, Integrating Curriculum, Assessment, and Teacher Professional Development: Singapore and the United States promised to go beyond the “popular but oversimplified views of Singapore’s successes.” The researchers at SRI repeated that merely adopting Singapore’s textbooks will not be enough to guarantee success here in the United States. It is a wide-spread, but mistaken belief, that a school can simply buy Singapore’s textbooks and duplicate their success in mathematics.

Specifically, they found that it wouldn’t be too challenging to duplicate parts of the Singapore Mathematics Syllabus in the United States, including:

• Creating standards that align to a common assessment.
• A guiding document like Singapore’s pentagon. Dr. Roschelle proposed that the U.S. could use Adding it up: Helping children learn mathematics. At over 450 pages, it doesn’t have quite the allure of the Singaporean document.
• Visual models. Singapore uses a concrete-pictorial-abstract methodology incorporating the model method and the U.S. has the Geometer’s Sketchpad.

Dr. Roshelle listed several items that would be somewhat harder to bring to the United States:

• A high level of professionalism among teachers.
• Systemic rotations through positions (at NIE, Ministry of Education and classrooms).
• Allowing teacher input at all levels of education.
• Systemic leadership.
• A culture of high expectations.

I’m starting with that culture of high expectations. Every school, every student.